Chapter 8

Chapter 8 Section 6

Objectives

1

Copyright © 2012, 2008, 2004 Pearson Education, Inc.

Solving Equations with Radicals

Solve radical equations having square root radicals.

Identify equations with no solutions.

Solve equations by squaring a binomial.

Solve radical equations having cube root radicals.

8.6

2

3

4


Solving Equations with Radicals.

A radical equation is an equation having a variable in the radicand, such as

1 3x or 3 8 9x x

Slide 8.6-3


Objective 1


Slide 8.6-4


To solve radical equations having square root radicals, we need a new property, called the squaring property of equality.

Be very careful with the squaring property: Using this property can give a new equation with more solutions than the original equation has. Because of this possibility, checking is an essential part of the process. All proposed solutions from the squared equation must be checked in the original equation.


Squaring Property of Equality

If each side of a given equation is squared, then all solutions of the original equation are among the solutions of the squared equation.

Slide 8.6-5


Solve.

Solution:

It is important to note that even though the algebraic work may be done perfectly, the answer produced may not make the original equation true.

9 4x

229 4x

9 16x 9 169 9x

7x 7x 7

Slide 8.6-6

EXAMPLE 1 Using the Squaring Property of Equality


Solve.

Solution:

3 9 2x x

2 2

3 9 2x x

3 9 4x x

3 33 9 4xx x x

9x

9

Slide 8.6-7

EXAMPLE 2 Using the Squaring Property with a Radical on Each Side


Objective 2

Identify equations with no solutions.

Slide 8.6-8


Solution:

Solve.

4x

2 2

4x

16x

16 44 4

4x

False

Because represents the principal or nonnegative square root of x in Example 3, we might have seen immediately that there is no solution.

x

Check:

Slide 8.6-9

EXAMPLE 3 Using the Squaring Property When One Side Is Negative


Solving a Radical EquationStep 1 Isolate a radical. Arrange the terms so that a radical is

isolated on one side of the equation.

Solving a Radical Equation.

Step 6 Check all proposed solutions in the original equation.

Step 5 Solve the equation. Find all proposed solutions.

Step 4 Repeat Steps 1-3 if there is still a term with a radical.

Step 3 Combine like terms.

Step 2 Square both sides.

Slide 8.6-10


Solution:

Solve 2 4 16.x x x

22 2 4 16x x x

2 22 24 16x xx x x

44 40 16xx x 4 1

4 4

6x

4x

Since x must be a positive number the solution set is Ø.

Slide 8.6-11

EXAMPLE 4 Using the Squaring Property with a Quadratic Expression


Objective 3


Slide 8.6-12


Solve

Solution:

2 1 10 9.x x

222 1 10 9x x

2 10 94 4 1 10 99 10x x xx x 24 14 8 0x x

2 1 2 8 0x x

2 8 0x 2 1 0x 4x 1

2x

Since x must be positive the solution set is {4}.

or

Slide 8.6-13

EXAMPLE 5 Using the Squaring Property when One Side Has Two Terms


Solve.

Solution:

25 6x x

625 66x x

2 2

25 6x x 225 12 325 256x x xx x 20 13 36x x

0 4 9x x 0 9x 0 4x

9x 4x

The solution set is {4,9}.

or

Slide 8.6-14

EXAMPLE 6 Rewriting an Equation before Using the Squaring Property



Errors often occur when both sides of an equation are squared. For instance, when both sides of

are squared, the entire binomial 2x + 1 must be squared to get 4x2 + 4x + 1. It is incorrect to square the 2x and the 1 separately to get 4x2 + 1.

9 2 1x x

Slide 8.6-15


Solve.

Solution:

1 4 1x x

1 1 4x x

2 2

1 1 4x x

1 1 2 4 4x x x

224 2 4x

16 4 16x 32

4 4

4x

8x The solution set is {8}.

Slide 8.6-16

EXAMPLE 7 Using the Squaring Property Twice


Objective 4


Slide 8.6-17



We can extend the concept of raising both sides of an equation to a power in order to solve radical equations with cube roots.

Slide 8.6-18


Solve each equation.

Solution:

3 37 4 2x x 3 2 3 26 27x x

3 3

3 2 3 26 27x x 2 26 27x x

20 26 27x 0 27 1x x

0 27x 0 1x 27x 1x

3 33 37 4 2x x

7 4 2x x 3 2

3 3

x

2

3x

2

3

27,1

or

Slide 8.6-19

EXAMPLE 8 Solving Equations with Cube Root Radicals

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